1. Simple Linear Regression
With Thanks to My Students in AMS572 – Data Analysis

2. 1. Introduction Example:
Brad Pitt: m
Angelina Jolie: 1.70m
George Bush :1.81m
Laura Bush: ?
David Beckham: m
Victoria Beckham: 1.68m
● To predict height of the wife in a couple, based on the husband’s height
Response (out come or dependent) variable (Y): height of the wife
Predictor (explanatory or independent) variable (X): height of the husband

3. History: Regression analysis:
●　 regression analysis is a statistical methodology to estimate the relationship of a response variable to a set of predictor variable.
●　when there is just one predictor variable, we will use simple linear regression. When there are two or more predictor variables, we use multiple linear regression.
● when it is not clear which variable represents a response and which is a predictor, correlation analysis is used to study the strength of the relationship
History:
● The earliest form of linear regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809.
● The method was extended by Francis Galton in the 19th century to describe a biological phenomenon.
● This work was extended by Karl Pearson and Udny Yule to a more general statistical context around 20th century.

4. A probabilistic model
We denote the n observed values of the predictor variable x as
We denote the corresponding observed values of the response variable Y as

5. Notations of the simple linear Regression
- Observed value of the random variable Yi depends on xi
- random error with
unknown mean of Yi
True Regression Line
Unknown Slope
Unknown Intercept

7. 4 BASIC ASSUMPTIONS – for statistical inference
Linear function of the predictor variable
Have a common variance,
Same for all values of x.
Normally distributed
Independent

8. Conditional expectation of Y given X = x
Comments:
1. Linear not in x
But in the parameters and
Example:
linear, logx = x*
2. Predictor variable is not set as predetermined fixed values,
is random along with Y. The model can be considered as a conditional model
Example: Height and Weight of the children.
Height (X) – given
Weight (Y) – predict
Conditional expectation of Y given X = x

9. 2. Fitting the Simple Linear Regression Model
2.1 Least Squares (LS) Fit

10. Example 10. 1 (Tires Tread Wear vs. Mileage: Scatter Plot
Example 10.1 (Tires Tread Wear vs. Mileage: Scatter Plot. From: Statistics and Data Analysis; Tamhane and Dunlop; Prentice Hall. )

12. One way to find the LS estimate and
The “best” fitting straight line in the sense of minimizing Q: LS estimate
One way to find the LS estimate and
Setting these partial derivatives equal to zero and simplifying, we get

13. Solve the equations and we get

14. To simplify, we introduce
The resulting equation is known as the least squares line, which is an estimate of the true regression line.

15. Example 10.2 (Tire Tread vs. Mileage: LS Line Fit)
Find the equation of the line for the tire tread wear data from Table10.1,we have
and n=9.From these we calculate

16. The slope and intercept estimates are
Therefore, the equation of the LS line is
Conclusion: there is a loss of mils in the tire groove depth for every 1000 miles of driving.
Given a particular
We can find
Which means the mean groove depth for all tires driven for 25,000miles is estimated to be miles.

17. 2.2 Goodness of Fit of the LS Line
Coefficient of Determination and Correlation
The residuals:
are used to evaluate the goodness of fit of the LS line.

18. We define:
Note: total sum of squares (SST)
Regression sum of squares (SSR)
Error sum of squares (SSE)
is called the coefficient of determination

19. Example 10. 3 (Tire Tread Wear vs
Example 10.3 (Tire Tread Wear vs. Mileage: Coefficient of Determination and Correlation
For the tire tread wear data, calculate using the result s from example 10.2 We have
Next calculate
Therefore
The Pearson correlation is
where the sign of r follows from the sign of since 95.3% of the variation in tread wear is accounted for by linear regression on mileage, the relationship between the two is strongly linear with a negative slope.

20. The Maximum Likelihood Estimators (MLE)
Consider the linear model:
where is drawn from a normal population with mean 0 and standard deviation σ, the likelihood function for Y is:
Thus, the log-likelihood for the data is:

21. The MLE Estimators Solving
We obtain the MLEs of the three unknown model parameters
The MLEs of the model parameters a and b are the same as the LSEs – both unbiased
The MLE of the error variance, however, is biased:

22. 2.3 An Unbiased Estimator of s2
An unbiased estimate of is given by
Example 10.4(Tire Tread Wear Vs. Mileage: Estimate of
Find the estimate of for the tread wear data using the results from Example 10.3 We have SSE= and n-2=7,therefore
Which has 7 d.f. The estimate of is miles.

23. Under the normal error assumption * Point estimators:
3. Statistical Inference on b0 and b1
Under the normal error assumption
* Point estimators:
* Sampling distributions of and :
For mathematical derivations, please refer to the Tamhane and Dunlop text book, P331.

24. * Pivotal Quantities (P.Q.’s):
Statistical Inference on b0 and b1 , Con’t
* Pivotal Quantities (P.Q.’s):
* Confidence Intervals (CI’s):

25. Statistical Inference on b0 and b1 , Con’t
* Hypothesis tests:
-- Test statistics:
-- At the significance level , we reject in
favor of if and only if (iff)
-- The first test is used to show whether there is a linear relationship between x and y

26. -- a sum of squares divided by its d.f.
Analysis of Variance (ANOVA), Con’t
Mean Square:
-- a sum of squares divided by its d.f.

27. Analysis of Variance (ANOVA)
ANOVA Table
Example:
Source of Variation
(Source)
Sum of Squares
(SS)
Degrees of Freedom
(d.f.)
Mean Square
(MS) F
Regression
Error
SSR
SSE 1
n - 2
Total
SST
n - 1
Source SS
d.f. MS F
Regression
Error
50,887.20 1 7
361.25
140.71
Total
53,418.73 8

28. 4. Regression Diagnostics 4.1 Checking for Model Assumptions
Checking for Linearity
Checking for Constant Variance
Checking for Normality
Checking for Independence

29. Checking for Linearity
Xi =Mileage Y=β0 + β1 x
Yi =Groove Depth ^ ^ ^
^ Y=β0 + β1 x
Yi =fitted value ^
ei =residual Residual = ei = Yi- Yi i Xi Yi ^ ei 1
394.33
360.64
33.69 2 4
329.50
331.51
-2.01 3 8
291.00
302.39
-11.39 12
255.17
273.27
-18.10 5 16
229.33
244.15
-14.82 6 20
204.83
215.02
-10.19 7 24
179.00
185.90
-6.90 28
163.83
156.78
7.05 9 32
150.33
127.66
22.67

30. Checking for Normality

31. Checking for Constant Variance
Var(Y) is not constant A sample residual plots when
Var(Y) is constant.

32. Checking for Independence
Does not apply for Simple Linear Regression Model
Only apply for time series data

33. 4.2 Checking for Outliers & Influential Observations
What is OUTLIER
Why checking for outliers is important
Mathematical definition
How to deal with them

34. 4.2-A. Intro Recall Box and Whiskers Plot (Chapter 4 of T&D)
Where (mild) OUTLIER is defined as any observations that lies outside of Q1-(1.5*IQR) and Q3+(1.5*IQR) (Interquartile range, IQR = Q3 − Q1)
(Extreme) OUTLIER as that lies outside of Q1-(3*IQR) and Q3+(3*IQR)
Observation "far away" from the rest of the data

35. 4.2-B. Why are outliers a problem?
May indicate a sample peculiarity or a data entry error or other problem ;
Regression coefficients estimated that minimize the Sum of Squares for Error (SSE) are very sensitive to outliers >>Bias or distortion of estimates;
Any statistical test based on sample means and variances can be distorted In the presence of outliers >>Distortion of p-values;
Faulty conclusions.
Example:
( Estimators not sensitive to outliers are said to be robust )
Sorted Data
Median
Mean
Variance
95% CI for mean
Real Data 5
6.0
20.6
[0.45, 11.55]
Data with Error
27.6
2676.8
[ ,91.83]

36. 4.2-C. Mathematical Definition
Outlier
The standardized residual is given by
If |ei*|>2, then the corresponding observation may be regarded an outlier.
Example: (Tire Tread Wear vs. Mileage)
STUDENTIZED RESIDUAL: a type of standardized residual calculated with the current observation deleted from the analysis.
The LS fit can be excessively influenced by observation that is not necessarily an outlier as defined above. i 1 2 3 4 5 6 7 8 9
ei*
2.25
-0.12
-0.66
-1.02
-0.83
-0.57
-0.40
0.43
1.51

37. 4.2-C. Mathematical Definition
Influential Observation
Observation with extreme x-value, y-value, or both.
On average hii is (k+1)/n, regard any hii>2(k+1)/n as high leverage;
If xi deviates greatly from mean x, then hii is large;
Standardized residual will be large for a high leverage observation;
Influence can be thought of as the product of leverage and outlierness.
Example: (Observation is influential/ high leverage, but not an outlier)
eg.1 with without eg.2 scatter plot residual plot

38. 4.2-C. SAS code of the tire example
Data tire;
Input x y;
Datalines; … ;
Run;
proc reg data=tire;
model y=x;
output out=resid rstudent=r h=lev cookd=cd
dffits=dffit;
proc print data=resid;
where abs(r)>=2 or lev>(4/9) or cd>(4/9) or
abs(dffit)>(2*sqrt(1/9));
run;

39. 4.2-C. SAS output of the tire example

40. 4.2-D. How to deal with Outliers & Influential Observations
Investigate (Data errors? Rare events? Can be corrected?)
Ways to accommodate outliers
Non Parametric Methods (robust to outliers)
Data Transformations
Deletion (or report model results both with and without the outliers or influential observations to see how much they change)

41. 4.3 Data Transformations Reason To achieve linearity
To achieve homogeneity of variance
To achieve normality or symmetry about the regression equation

42. Types of Transformation
Linearzing Transformation
transformation of a response variable, or predicted variable, or both, which produces an approximate linear relationship between variables.
Variance Stabilizing Transformation
make transformation if the constant variance assumption is violated

43. Linearizing Transformation
Use mathematical operation, e.g. square root, power, log, exponential, etc.
Only one variable needs to be transformed in the simple linear regression.
Which one? Predictor or Response? Why?

44. e.g. We take a log transformation on Y = a exp (-bx) <=> log Y = log a - b xXi Yi ^
log Yi
Y =
exp (logYi) Ei
394.33
5.926
374.64
19.69 4
329.50
5.807
332.58
-3.08 8
291.00
5.688
295.24
-4.24 12
255.17
5.569
262.09
-6.92 16
229.33
5.450
232.67
-3.34 20
204.83
5.331
206.54
-1.71 24
179.00
5.211
183.36
-4.36 28
163.83
5.092
162.77
1.06 32
150.33
4.973
144.50
5.83

45. Variance Stabilizing Transformation
Delta method : Two terms Taylor-series approximations
Var( h(Y)) ≈ [h(m)]2 g2 (m) where Var(Y) = g2(m), E(Y) = m
set [h’(m)]2 g2 (m) = 1
h’(m) =
h(m) =    h(y) =
e.g. Var(Y) = c2 m2 , where c > 0, g(m) = cm ↔ g(y) = cy
h(y) = = =
Therefore it is the logarithmic transformation

46. 5. Correlation Analysis
Pearson Product Moment Correlation: a measurement of how closely two variables share a linear relationship.
Useful when it is not possible to determine which variable is the predictor and which is the response.
Health vs wealth. Which is predictor? Which is response?

47. Statistical Inference on the Correlation Coefficient ρ
We can derive a test on the correlation coefficient in the same way that we have been doing in class.
Assumptions
X, Y are from the bivariate normal distribution
Start with point estimator
r: sample correlation coefficient: estimator of the population correlation coefficient ρ
Get the pivotal quantity
The distribution of r is quite complicated
T0: test statistic for ρ = 0
Do we know everything about the p.q.?
Yes: T ~ tn-2 under H0 : ρ=0

48. Bivariate Normal Distribution
pdf:
Properties
μ1, μ2 means for X, Y
σ12, σ22 variances for X, Y
ρ the correlation coeff between X, Y

49. Derivation of T0
Therefore, we can use t as a statistic for testing against the null hypothesis H0: β1=0
Equivalently, we can test against H0: ρ=0

50. Exact Statistical Inference on ρ
Example
A researcher wants to determine if two test instruments give similar results. The two test instruments are administered to a sample of 15 students. The correlation coefficient between the two sets of scores is found to be Is this correlation statistically significant at the .01 level?
H0 : ρ=0 , Ha : ρ≠0
for α = .01, = t0 > t13, .005 = 3.012
▲ Reject H0
Test
H0 : ρ=0 , Ha : ρ≠0
Test statistic:
Reject H0 iff

51. Approximate Statistical Inference on ρ
There is no exact method of testing ρ vs an arbitrary ρ0
Distribution of R is very complicated
T0 ~ tn-2 only when ρ = 0
To test ρ vs an arbitrary ρ0 one can use Fisher’s transformation
Therefore, let

52. Approximate Statistical Inference on ρ
Sample estimate:
Z test statistic:
CI for ρ:
We reject H0 if |z0| > zα/2

53. Approximate Statistical Inference on ρ using SAS
Code:
Output:

54. Pitfalls of Regression and Correlation Analysis
Correlation and causation
Ticks cause good health
Coincidental data
Sun spots and republicans
Lurking variables
Church, suicide, population
Restricted range
Local, global linearity

55. Summary Linear regression analysis Model Assumptions Correlation
Coefficient r
The Least squares (LS) estimates: b0 and b1
Probabilistic model
for Linear regression:
Correlation
Analysis
Outliers?
Influential Observations?
Data Transformations?
Confidence Interval & Prediction interval

56. Sample correlation coefficient r
Least Squares (LS) Fit
Sample correlation coefficient r
Statistical inference on ß0 & ß1
Prediction Interval
Model Assumptions
Correlation Analysis
Linearity Constant Variance Normality Independence

57. Questions?